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THE UNIVERSITY 
OF ILLINOIS 
LIBRARY 


|) From the collection of 
» Julius Doerner, Chicago 
Purchased, 1918, 


SB+O-84 513.2 
Gols | 


MATHEMATICS UBRARY 


¢' 
e 

1 
i 


fo: VV. GOODHUPF, 


CHICAGO. 


Entered according to Act of Congress, in the year 1885, 
By 8S. W. GOODHUE, 
in the Office of the Librarian of Congress, at Washington, D. C. 


CHICAGO: 
Printed by H. H. HOFFMANN & Co. 


MATHEMATICS LIGRARL 


INTRODUCTION. 


1. Addition, Subtraction, Multiplication and 
Division are commonly termed the four fundamental 
principles of Arithmetic. Operations in Addition, 
Subtraction and Division require a certain amount 
of mental effort which cannot be materially dimin- 
ished by any known labor-saving appliances. The 
expert arithmetician may shorten the time, but he 
cannot escape the labor incident to these processes. 


2. But Multiplication is an abbreviated form of 
Addition, the distinction being that by one process 
we laboriously ascertain the sum of several unlike 
numbers, while by the other we more easily compute 
the sum of many like numbers. As in some 
branches of industry, one man aided by machinery 
may accomplish the work of 10 men, or may abridge 
the time, as well as improve the quality of his own 
labor in like proportion; so in Multiplication, a 
skillful use of mental appliances may effect a corres- 
ponding economy of time and effort. 


01425 


4 INTRODUCTION. 


3. These points may be illustrated by two 
examples; the first showing the time-saving value ~ 
of expertness; the second, the labor-saving efficacy 


of abbreviation. 


I. A tailor sews upon 13 garments, the 
number of buttons expressed by the mar- 
ginal figures: 

How many buttons in all ? 

Solomon Slowbones points with his finger 
and drawls with his mouth, thus: “s-e-v-e-n 
a-n-d o-n-e a-r-e e-i-g-h-t, a-n-d f-o-u-r 


a-r-e t-w-e-l-v-e,” etc., while Ed. Expert, 


BD DO RH KR DDR BH BH 
— 


Hm Co bo bo ® WW Ol WwW > bo 
ee eee ae ee TU 


pi 
~] 


holds his tongue, groups several figures 
mentally, and thinks nimbly, 12, 22, 38, 48, ete., 
or taking both columns at once, 38, 78, 112, 148, 
171, 208. 


II. A tailor sews upon 13 coats, 16 buttons 
each: How many buttons in all? This, as a 
problem in Addition, requires the writing of the 
number 16 thirteen times; but by the abbreviated 
process of Multiplication as commonly practiced, 16 
is written once with 13 under it—4 figures instead 
of 26—and then brain and hand work together, as 
shown on the following page. 


INTRODUCTION. 5 


1 6 
1 
Brain work 3x6=18 Hand work 8 
exe to 3+1—4 
£6 6 6 
ag Poe | 1 
8 8 
6+4=10 0 
I+1= 2 2 


Ed. Expert, abbreviating the abbreviation, with 
less brain work better directed and with no hand 
work at all, thinks, as the lightning flashes, 


. 19 tens and 6 threes, 208. 


Has he multiplied 16 by 13? By no means. 
He is exempted from so hard a task by his famil- 
iarity with the relations of numbers. He has simply 
apprehended 1910 as a relative approximation to 
16 13, and 63 as the difference to be added. 

4. With almost equal facility, in each case 
substituting for a difficult computation, 


AN EASY COMPUTATION AND A DIFFERENCE, 


he solves the following problems: 


6 INTRODUCTION. 


1. 17x19 8x 40 +3 323 
R27 x27 17x 40 +49 729 
3. 29x84 50x50 —64 2436 
1 §382x86 32x 100 —8xX56 2752 
" 13286 ¥% x 8600(2867) —115 2752 
5. 36X38 40x40+-8 = 9407 ee 
6. 44x48 1546x100 oe 2113 
7. 57X64 60X61 i 3648 
8. 64x64 35+4x100 +196 4096 
9. 73x89 62 x 100 +297 6497 
10. 263X278 270270 4-214) 7S 


5. For all factors under 100, and for multi- 
tudes of greater ones, the 


GEOMETRICAL BIRD METHOD 


of multiplication, as developed in this work, fur- 
nishes the average accountant, if not the average 
school-boy, a system of simple and intelligible data 
for various mental formulas, easy, rapid and accu- 
rate; an insight into which will release him at once 
and forever from the slow, laborious and often in- 
correct drudgery of the ordinary written process. 


6. By the geometrical diagrams with which the 
work is illustrated, integrity of process, and accu- 


INTRODUCTION. ~ 7 


racy of result, are infallibly demonstrated,—the 
mind being, as it were, electrified by a broad scien- 
tific comprehension of the nature and scope of the 
problem, as a wHoxz, instead of being confused and 
lost in the multiplicity of its parts. 


7. For example, with so simple a problem as 
13X16, the ordinary cipherer is seldom quite sure 
of his correctness. His computation has many 
parts, as a tub has many staves. The tub, as a 
whole, is an incomprehensibility. He can only 
apprehend one staveatatime. So, fearing a hidden 
leak, he carefully examines stave by stave, and joint 
by joint, and finds only a negative assurance of 
soundness in the lack of a positive discovery of 
defect. He has no comprehensive idea of what the 
product of 13X16 ought to be. 


8. Let him now inspect this 3 
diagram, representing the space 


occupied by 13 rows of 16 but- 19 
tons each, separated by crossed _ 
threads into 4 parts, viz: 

10 tens 


3 tens + and 6 threes. 
6 tens 


8 INTRODUCTION. 


Without ciphering, he may now perceive just 
what the product of 13x16 must be. His tub is 
glass, and through its transparency gleams its 
infallibility. 

9. The general divisions of the subject are as 


follows :— 


I. The Multiplication Table, considered, 1st, 
in its superficial aspect, as a collection of 144 simple 
products to be memorized in childhood, and 2nd, 
in its deeper significance, as an agency in the devel- 
opment of countless larger products, by virtue of 
the geometrical position, or the mechanical manipu- 
lation of the figures of which it is composed. 


II. Multiplication, as a language, its letters, 
syllables and product-words. Slow and laborious 
writing and spelling backwards, carrying and blunder- 
ing, compared with quick, easy and demonstrably 


accurate reading forwards. 


Il. 99x99, or mental combination of several 
memorized partial products as syllables of a product- 
word—symbolized by the body, head and wings of 
of the Geometrical Bird. 


IV. Written Multiplication, or manual combi- 
nations of partial products, intelligibly arranged in 


INTRODUCTION. 9 


the Diamond or Pyramid form. In this system, 
partial products being written entire, there is 


NOTHING TO CARRY, 


(except in the final addition of partial products), 
and consequently, a most prolific source of confu- 


sion, and error is effectually avoided. 


10 THE MULTIPLICATION TABLE. 


THE MULTIPLICATION TABLE. 


10. A small child is taught to say, and to re- 
member the saying, 


3 times 5 are 15. 


A parrot learns the same lesson, and then brute and 
human are equal. Hach repeats a form of sound 
without sense, of words without meaning, and one 
knows as much of arithmetic as the other. 


11. The child takes another lesson—an object 
lesson, of fingers on three hands, or of buttons, 3 
on each of five gloves; and by actual count of the 
entire 15, there comes to his mind a comprehension 
of the meaning of the repeated words, a demonstra- 
tion of the verity of the remembered theorem. 
Relations are now changed. The boy has knowl- 
edge, while the bird has only learning. 


12. A few years later, the boy in partnership 
with a pencil, having learned the Multiplication 
Table and the Rule, is enabled, by writing 14 
figures and making as many minor computations, 
to cipher out the product of 99x99. He could no 


THE MULTIPLICATION TABLE. La 


more do this without the pencil than the pencil 
could do it without him. Now, the parrot has 
learning, and the boy more learning, but he 
lacks knowledge. If the correctness of his cipher- 
ing is questioned, he dare not affirm it, for he can- 
not demonstrate it; but doubtfully reviews the 
work in a blind hunt for errors. He knows not 
how many 99x99 ought to be. 


13. He takes a larger lesson in buttons, 100 
rows of 100 buttons each, arranged as a geometrical 
square. One row being taken from a side and one 
from an end, 99x99 buttons remain. How many 
have been removed ? Not 200, for the corner but- 
ton was common to both rows. Then 200 minus 1 
being taken from 100 hundred, there remain 98 
hundred plus 1, or 9801. No pencil is needed to 
make this computation; no external proof to verify 
it. Independent, transparent, self-evident, the tub 
stands squarely on its own bottom. No questioning 
can now shake the boy’s confidence. He knows 
what 99x99 must be; and again the child with 


knowledge is superior to the parrot with learning. 


14. There be multitudes of homo-parrots—school- 
boys, teachers, men of letters, and men of business 


12 THE MULTIPLICATION TABLE. 


—with abundant learning of the multiplication 
table and the rule; but with scant knowledge of the 
underlying principles of Multiplication: They need 
to count buttons and dissect geometrical birds. 


15. A child’s fingers are too few for larger 
demonstrations; and buttons in quantities are not 
conveniently portable—besides being noisy and 
troublesome” when dropping and _.rolling on the 
school-room floor—and so it was a happy thought 
of old Pythagoras, as tradition affirms, the con- 
struction of a geometrical table, the little squares of 
which, though convincing not the finger-tips like 
buttons, yet standing for buttons to the eye, may 
be counted, and thus convey to the child-mind a 
practical demonstration of each proposition, from 
1x 1=—1 up to the grand climax 12x 12—144, 


16. The beginning of knowledge, as distinet 
from learning, comes only from such actual count- 
ing of tangible objects representing small products. 
Later lessons may be based on immaterial buttons 


in imaginary rows.* 


* “Seemed washing his hands with invisible soap, 
In imperceptible water.”—Hood. 


THE MULTIPLICATION TABLE. 13 


17. Our historic world was thousands of years 
old before it saw the first Multiplication Table; and 
even now there exist many tribes of men who can- 
not count their own fingers and toes. A man, 
wondrous learned and wise above his fellows was 
he, whether Pythagoras or another, who invented 
the geometrical table. But imitators now need not 
his full measure of wisdom. Indeed the 144 squares 
of the table may be correctly filled by a person 
utterly ignorant of Multiplication, or even of Addi- 
tion. He has only to number consecutively the 
144 quarter-inch marks on a yard-stick, and then 
transfer to the first line of the table the first 12 
numbers; to the second line, the first 12 which 
occur at half-inch intervals, and so on to the twelfth 
line, the appropriate numbers for which are found 
three inches apart. 


18. Can these dry bones live? Can the soul 
of numbers vivity a dull geometrical. clod thus 
ignorantly conceived and mechanically developed ? 
Is there in this assemblage of figures any hidden 
thing beyond the 144 products which childhood 
cons by rote and crones in sing-song ? Investiga- 
tion proves that by virtue of their geometrical posi- 


14 THE MULTIPLICATION TABLE. 


tion alone, these soulless characters have marvel- 
lous potency in the solution of stupendous problems, 
invisible if not incomprehensible to the childish 
student.* 


19. Here, for example, is a miniature table in 
which the ordinary observer sees nothing beyond 
2x2—4. If we add together the upper numbers, 
1 and 2, and the side numbers 1 and 2—the number 
1 doing double duty—we have 3x3, the product of 
which equals the sum of all the num- 
bers in the table. If, instead of add- 
ing the factor figures, we consider 


them as expressing the number 12, 
then the product of 12x12 appears 
by turning the point a to the left and adding per- 
pendicularly. Ifthe point a be turned to the right 
and the reversed figures be read as 21X21, the 
product 441, appears by adding as before. If the 
point a be turned upward or downward, and the © 
factors be read as 12X21, the product 252 is simi- 
larly obtained. 


20. In the next table may be found the pro- 


* **There are more things in heaven and earth and the multipli- 
cation table, Horatio, than are dreampt of in your philosophy.”— 
Shakespeare & Co. 


THE MULTIPLICATION TABLE. 15 


duct of 1+2x1+2-+3, or 3x6=the sum of all 
the numbers in the table; also with « at the 


Left, 12x 123=—1476 
Bottom, 12 321=3852 
Right, 21 x 321=6741 

Top, 21 x 123=2583 

If the side factors 1 and 2 be taken as factors 
only, and not included in the addition of products, 


we have with a at the 


Left, 12 23=276 
Bottom, 12« 32—=384 
Right, 21 x 32=672 
Top, 21 « 23=483 


21. In this table may be found the products of 
4x 4=16 

13x 13=169 

13x 31=403 

31x 31=961 


With a cipher in each vacant square may be 
developed the products of 
103 x 103=10609 


301 x 103=31003 
301 301=90601 


16 THE MULTIPLICATION TABLE. 


22. In this section representing the four central 
squares of the table, under and at the right of the 
factors 6 and 7, the product of 


6+7x6+7, or 13X13 equals 3 
the sum of the four numbers, | 6 
169. Addition with the point 4 a3 
a turned to the left developes ) 9 
the product of- 67 x 67 = 4489. 

“ 

A change in the position of a a ‘ 
two figures, developes in the = E 
second diagram the product of - f 
607 x 607 = 368449. 2,;4 9 


23. This section represents the four squares at 


the right lower corner of the table, under and at 
the right of the factors 11 


; os Bl 10 

and 12. The sum of the 1 1 
four numbers 529 is the 9 3 
product of 11+12x11-+4 1 - 


12 or 23X23. The pro- 
duct of 1112 x 1112 — 
1236544 is obtained by 


addition with the point a 


turned to the left. The factor numbers 11 


ll and 12 overlapped as in the margin 12 
form by addition the number 122. 122 


THE MULTIPLICATION TABLE. 17 


The numbers in the “ l 
second diagram similarly 9 
overlapped, develop by ad- 
dition, the point a being pel hee 

3 4 


turned to the left, the pro- | 
duct of 122x 122—14884. 2 


24, Finally, the whole Multiplication Table 
may be regarded as a single problem, one factor 
being the sum of the numbers in the upper line, the 
other, of those in the left column, while the pro- 
duct is the sum of the entire 144 numbers. , This 
problem may seem at first sight a formidable one; 
but the elements of its solution are wonderfully 
simple. The upper line of the table may be seen to 
consist of six pairs of numbers, viz: 112, 211, 3 10, 
49,58and6 7; the sum of each pair being 13, 
the average of each number 63, and the sum total 
78. In the second line these proportions are 
doubled, and so on by a regular progression to the 
twelfth line. Conditions precisely similar appear if 
the table is scanned in columns. 


25. Itfollows that the square of 64 or 42+ must 
bethe average number for the entire table, and there- 
fore the product of 421 by 144 must be the sum 


18 THE MULTIPLICATION TABLE. 


total: also, that the same aggregate must result 
from squaring the number 78. By methods here- 
after explained, the square of 78 is mentally de- 
duced from the square of 75 or of 80—an easy 
computation and a difference—thus, 5634+450= 
6084, or 6404—320—6084. The problem 424 x 144 
divested of its fractional complication by substitut- 
ing the equivalent factors 169 x 36 is solved at sight, 
thus: 4854+1230—6084. 


26. As the number 169, the square of 13, repre- 
sents the average aggregate of four squares, the 
question naturally arises, are there in the table any 
four numbers of exactly that sum? Itis a curious 
fact that the entire table is made up of 36 such 
sroups of four—one odd number and three even 
ones in each group—all symmetrically arranged 
around a common center. The figures comprising 
each group are equi-distant from the center of the 
table, and also equi-distant from each other; and 
stand at the angles of a perfect geometrical square 
—unless the table itself varies from that form. 
The sum of a side of one of these squares is invari- 
ably a multiple of 13; and the product of the largest 
by the smallest number is in every case equal to 
that of the other two. 


THE MULTIPLICATION TABLE. 19 


These groups of 169 may be readily located by 
reference to the table (28), thus: 
1 12 2 24 Si Saplé 4 48 
12 144 Blot 10 120 9F108 


By analysis, the fourth group is seen to consist 
of 
4-1s and 4 12s 
wersesetand 9. 1¢ 


13 13 13 13 


MULTIPLICATION TABLE. 


1 
2! 4] 6] 8/10/12/14]16]18] 20] 29) 24 
3] 6] g/12/15/18/21|24|27] 30) 33) 36 
~4{ 8{12] 16/20 | 24/28 | 32 40; 44} 48! 
610] 15 | 20 | 95| 30/35 140145] 50) 55] 60 
6/12] 18 | 24| 30 | 36/42/48] 54] 60) 66] 72 
714] 21 | 28 |35 | 42] 49 | 56 | 63 | 70] 77] 84 
8 {16 | 24/32 | 40 | 48 | 56 | 64] 72] 80] 88] 96 


ee | | | | | es | | 


io) 
— 
So 
bo 
we 
(SX) 
3) 
eS 
es) 
pha 
ce 
On 
(=P) 
(one) 
as 


90 |{0Q | 110/120 


re) 


12 | 24 | 36 | 48 108 


120] 132/144 


20 THE MULTIPLICATION TABLE. 


27. The same principle prevails in any square 
multiplication table, whatever its magnitude, the 
sum of the group of four numbers being in any case 
equal to the next square number outside the table. 
In an odd-numbered table, as 5X5 or 9X9, the 
central figure, 9 or 25, represents the average num- 
ber, and is surrounded, the first by 6 groups of 36 
each, the second by 20 groups of 100 each. 


28. In the accompanying table of 12x12 
squares, heavy lines at 9x9 indicate the extent of a 
table of unit factors. Factors too large to be multi- 
plied mentally are necessarily subjected to a written 
process, by which several partial products are 
obtained and added together. For the numbers 
representing these partial products, the operator 
depends almost invariably, not on original compu- 
tations but on the remembered lessons of this 


lesser table. 


29. We may imagine a person unable, from 
some mental defect, to memorize these lessons and 
yet able by using this table for reference to obtain 
and set in order for addition the partial products of 
factors of any magnitude by means of an improvised 
geometrical table like the one here represented. In 


THE MULTIPLICATION TABLE. 21 


this table the product 
of 684 x 397 is obtained 
by placing one factor 
at the top, one at the 
left, the 9 partial pro- 
ducts within the 9 
squares, and their sum 
971,548, extended 


around the bottom and right of the table, the point 
a being uppermost during the addition. 


30. By the ordinary process, instead of 9, then 
are developed 3 partial products—which may be 
traced in the 3 lines of squares in this table, viz: 


7X 684 4788 
90 x 684 6156 
300 x 684 2052 
271548 


31. By either method, only the significant. 
figures of the partial products are written, the final 
ciphers being omitted. The full expression of the 
partial products in the table requires that the 
ciphers in each square be placed at the right of the 
significant figures. 


22 THE MULTIPLICATION TABLE. 


Thus there appear: 


1 product of units or ones, 28 

Dae Po tens 56, 36, 92 

ote hundreds, 42, 72,* 12 126 

EAS ry ten hundreds, 54, 24 . 78 

iss? hundred-hundreds 18 
271548 


32. An observable feature in the Multiplication 
Table (28), is the diagonal line of square numbers, 
1, 4, 9, 16, etc. A square number being the pro- 
duct of equal factors, has but one place in the table, 
but other products are duplicated by the inversion 
of their factors, the No. 132, for example, appearing 
both as the product of 11x12 and of 12x11. 
Hence the two sections of the table as divided by 
this line, are exact duplicates. From a table minus 
one section, a boy minus one eye might learn the 
144 products; but both sections as well as both . 
eyes are usually considered desirable. 


33. Diagonal lines parallel to that of square 
numbers, consist of numbers diminished by square 


numbers. Thus in 


5 lines opposite 36 are 35 32 27 20 I1 
the differ’ce from 36 being 1 4 Ee 


* 72is a product of ten-tens, identical with hundreds. 


THE MULTIPLICATION TABLE. 23 


- THE TABLE OF SQUARE NUMBERS. 


34. A method of Multiplication without multi- 
plying is based on the use of a reference table of 
square numbers, written or printed, combined with 
a process of Addition, Subtraction, and Division. 


35. Consecutive square numbers are identical 
with the sums of consecutive odd numbers. Hence 
a person ignorant of Multiplication may construct 
this table by a series of additions, thus: 


Numbers, | yi 5 4. 5 6 
Squares, 1+3—4+5—9+7=—16+9=25+11=36 

As the work advances, its correctness may be 
tested at every 10th number. Thus 4x4 being 16, 
40x40 must be 1600. To change 40x40 buttons 
to 4141 buttons, 2 sides and a corner must be 
covered or 2X40+1, making the square of 41=— 
1681. Then to change the square of 41 to that of 
42, the next odd number 83, must be added making 
1764. 


36. Square numbers, except the number 1, are 
either even, and multiples of 4, or odd, multiples of 
4 plus 1. The first 10 square numbers will be 
found on the following page. 


24 THE TABLE OF SQUARE NUMBERS. 


1 4,..9 16 25.36 49° G25 sie 


The terminal figures of these 10 are repeated in 
every succeeding 10. Thus if a number ends with 


0) its square ends with 00 
00 0000 
1 or 9 1 remove from 0 1 
2 or 8 2 * 4 
DOr. 3 . 9 
4 or 6 4 a 6 
5 5 A 25 


No square number ends with 2, 8, 3 or 7, or with 
an odd number of ciphers, or with 5 other than 25. 
These characteristics may be verified by the accom- 
panying table, which contains the squares of all 


numbers up to 99. 


25 


THE TABLE OF SQUARE NUMBERS. 
TABLE OF SQUARE NUMBERS, 


F8IG | LPOG 


VGLO | 19E9 


VHSE | LéELE 


FOLGE | 1096 


POLL | T89T 
FCOL | 196 
V8P LFF 
VFI LéT 
iy { 
G I 


v9FS | 18Eé8 


_————— 


O0OF9 


OO6F 


0096 


0066 


0091 


0018 06 


08 
OL 


09 


OG 


OF 
0€ 


26 THE TABLE OF SQUARE NUMBERS. 


37. The method of obtaining products by refer- 
ence to this table is based on the fact that the 
square of the sum of 2 factors includes the square 
of their difference, plus 4 times their product. 


The process is as follows: 


I. Find the square of the sum of the factors. 
II. Find the square of their difference. 
111. Subtract one from the other. 
IV. Divide the remainder by 4 


Example I. 9x3. 


9+3=12 Square of 12, 144 
9—3=—6 Square of 6, 36 
Remainder, 108 
4 of remainder, 27 


38. Demonstration. If 4 cards of buttons, 
3x9 on each card, be 


placed together as repre- | a ae 
sented in the accompany- 
ing diagram, they form, 6 
1. An outer square, 
measured on each side by 
12 buttons. 


2. An inner vacant 


square, measured on each side by 6 buttons. 


THE TABLE OF QUARTER-SQUARES. 27 


Example I. 57 x 38. 


57+38—95. Square of 95, 9025 
57—38=19. Square of 19, 361 
Remainder, 8664 
3 of remainder, 2166 


TABLE OF QUARTER-SQUARES. 


39. The necessity for a final division by 4, in 
each operation, has been obviated by dividing the 
entire table by 4, or in other words, constructing a 
table of quarter-squares. 


40. The quarter-square of any number is equal 
to the square of half that number. Thus the 
quarter square of 12 is the square of 6, or 36; of 
13, the square of 64 or 424. In this system, how- 
ever, the fraction + pertaining to the quarter-squares 
of odd numbers is dropped without affecting the 
result. The proximate quarter-square is the pro- 
duct of the two numbers proximate to the half, as 
of 13, 6X 7=42. 

41. This table, like the table of square num- 
bers, may be constructed by successive additions ; 
the consecutive quarter-squares of even numbers 
being identical with the sums of consecutive odd 


numbers, and vice versa, per example on next page. 


28 THE TABLE OF QUARTER-SQUARES. 


ven Nos. , 4 Bose 10 ‘12 
4 Squares, 143= 4+5=9+7=1649=25+11=36 
Odd Nos, 3 5 i 9 11 13 


a Syrs., 2+4—6+6=12+8=20+10=—30+12=42 
42. By reference to a table of quarter-squares, 
products are obtained as follows: 


x, 1. 9X 3. 
9+3=12. Quarter-square, 36 
9—3—6. 7 9 
Remainder, 27 
Ex. IT. 57 X 38. 
57+38=—95. Quarter-square, 2256 
57—38=—19. 2 90 
Remainder, 2166 


43. This method has the following earnest com- 
mendation, in “Sane’s Hicguer Aritumetic.” Kd. 
and Lond, 1857: 


“M. Anton Voisin published in 1817 a table of 
quarter-squares of all numbers up to 20000, which 
enables us to find the product of any two numbers 
up to 1000. Mr. Laundy has lately published a 
table of quarter-squares up to 100000. This work 
should be in the hands of every professional cal- 
culator.”’ 

Popular judgment, however, has consigned the 
system to oblivion, from which it is here resurrected 
in compliment to its ingeniousness rather than to 
its usefulness. 


MULTIPLICATION AS A LANGUAGE. 29 


MULTIPLICATION AS A LANGUAGE. 


44, Multiplication may be considered as a lan- 
suage, whereof the numerals are letters; the par- 
tial products, syllables, and the sum of the sylla- 
bles, product-words. 


45. The alphabet of this language consists of 
the 10 numerals, 


0 1 2 3 + 5 6 7 8 4 


commonly termed the cipher and the 9 digits. The 
digits are also called significant figures. 


46. A syllable is the product of two numbers, 
as 9X9, or any lesser factors. No larger syllables 
than those contained in the lesser Multiplication 
table, (28) occur in the ordinary written process ; 
but in mental computation, any product may be 
treated as a syllable when recalled by an act of 
memory, or when instantly computable, as of 12x 
12, 15x18, 25x 25, 75 x 64. 


47. Letters or syllables followed by ciphers 
assume magnified proportions, but are not thereby 
rendered more difficult of computation, Thus in the 


30 SPELLING BACKWARDS. 


formula 80x 1200=96000, the 3 ciphers from the 2 
factors mechanically supplement the simple product 
of 8x12, just as 3 dots are made to complete the 
writing of the word Division. 


48. In any product word the number of sylla- 
bles equals the product of the number of digits in 
one factor by the number in the other, thus: 2, 3, 
or 4 digits in each factor produce respectively 4, 9, 
or 16 syllables; while 2, 3, or 4 digits multiplied 
by 1 digit produce 2, 3, or 4 syllables. 


es ae 


SPELLING BACKWARDS. 


49. The word ‘‘ Multiplication”’ requires for 
its writing, some 50 distinct movements of pen or 
pencil, for its spelling, 14 letters; for its pro- 
nouncing, 5 syllables; but for its silent reading, 
only the passing glance of a practiced eye. 


50. The various methods of Multiplication, as 
applied to factors under 100; for example, 46 x 48, 
exhibit contrasts equally striking. The ordinary 
computer, ignorant of syllables, takes the slowest 


CARRYING AND BLUNDERING. 31 


and most toilsonie method, wasting time and 
energy on 50 manual movements in 14 figures and 
2 lines, meanwhile making 10 computations—thus 
laboriously writing backwards with many letters a 
word which may be instantly read forwards with 
2 syllables, thus 44x50+4x2. When all is done 
it is only half done. There are 24 chances of error 
in his 14 figures and 10 computations, and he 
seldom ventures to omit reviewing the entire work 
in search of the possible mistake. 


ep eres iSeeate 


CARRYING AND BLUNDERING. 


51. The written process prescribed in the “Rule’’ 
of the school-books, although theoretically sound 
and philosophically correct, is practically the occas- 
sion of infinite blundering. While sufficiently 
simple to be apprehended and practiced by children 
7 years of age, it is yet so complex as hardly to be 
comprehended by the same children when that age 
is doubled. 

“ Likewise ye Monkie, albeit he turneth about 
y. Crank of ye Hande-Organ, hath natheless Small 
Knowledge of ye Musickal Product  thereof.’’— 
Antique Author. 


32 CARRYING AND BLUNDERING. 


52. The hidden spring of this defect is the 
ignoring and the unconscious mutilation of the 
natural syllables of a product. A child, unable to 
read the word, Multiplication, may yet succeed with 
- the syllables, Mul-ti-pli-ca-tion, and again be sorely 
puzzled by the no-syllables, Mu-lt-ip-li-cat-ion. So 
a child may comprehend the 9 syllables of the fac- 
tors 684 « 397, either within or without the lines of 
a geometrical table (29, 117); and may further 
apprehend the fact that each syllable is unques- 
tionably right and in its right place; and thata 
correct addition only is required to insure a correct 
product. | 


53. Turning from these 9 simple syllables, all 
verified by the multiplication table, to the three 
complex partial products (30); each consisting of 
syllables butchered as soon as born—the tens 
chopped off from one and stuck on to another—what 
can the child make of them? They may be right, 
if the ciphering is right ; as the tune may be ground 
out if the crank is properly turned; but neither child 
nor monkey can distinguish the right from the 
wrong. 


54. The school-boy, by the time he reaches the 
‘‘first class,” is so thoroughly satwrated with the 


CARRYING AND BLUNDERING. 33 


Addition and Multiplication tables, that he cannot 
miss the resulting syllable of 8-+-8 or 4x4. He 
fails, not in setting down the 6, but in carrying the 1. 


55. The device of carryine, the child’s riddle, 
and the man’s life-long stumbling-block, is the 
source of nine-tenths of the every-day blundering 
in Multiplication. 


56. If the world were restricted to one tool, it 
should be the jack-knife, which fits the universal 
pocket of boy or man, and is pre-eminently the 
implement of all work; if to one method of Multi- 
plication, the old one, with its little and big blades, 
the ‘‘table’’ and the ‘‘ rule,”’ has been fitted to the 
universal head, and like the ox-team, is to be com- 
mended for the slow and heavy labor it has accom- 


plished in the past. 


57. But, the jack-knife being too delicate for 
the ship-carpenter and too clumsy for the surgeon, 
the broad-ax and the lancet have been evolved; and 
the ox-express is displaced by steam and electricity. 
So for the practical computer the Multiplication 
table is too short, the old written process too long 
and laborious. The child’s table, 12x12, should 
be supplemented by the man’s table, 99x99, not 


34 LIGHTNING CALCULATORS. 


written, not memorized, but instantly computable; 
and the written process—never applied to factors 
under !00--shortened, simplified and freed: from 
the errors incident to carrying, should partake more 
of the electric alertness of the brain, less of the 
dull plodding of the hand. 


LIGHTNING CALCULATORS. 


58. Partly in this direction, and partly oppo- 
site thereto, is the method of the ‘lightning caleu- 
lator’’; combining on the one hand more head- 
work with less hand-work, and on the other greater 
speed with less simplicity, By this method the 
product is written directly in one line, without the 


intervention of partial products. 


Ex. 684 x 397. 684 
397 
Written, 271548 
Performed—the numbers carried being expressed 
in Italics : — 
7x4 28 
7X8 and 9x4 +2 94 
7x6 and9x8and3x4 +49 135 
9x6 and 3x8 +13 91 


3X6 +9 27 


LIGHTNING CALCULATORS. 35 


59. The student may be assisted in compre- 
hending this process by practicing the following ex- 
amples,in which the digits correspond to the several 
orders of units numbered from right to left. 


i Dees 321 X 321. 


Ex. IT. 4321 x 4321. 
Ex. WI. 54321 54321. 


Ex. I is written, _ 321 
321 
103041 
Performed: 
1x1 1 (1 product of units. 
1x2 and 2x1. 4 (2 products of tens. 
1X3 and 2X2 and 3X1 IO (3 “ hundreds. 
2x3 and 3x2 +] 13 (a? thousds. 
brag +i 10 (1 “ 10 thousds, 


60. This operation may be made partly mechan- 
ical, by writing one of the factors with the digits 
transposed, thus: 123 on a moveable slip of 
paper and passing it from right to left under. the 
other factor, taking the products, or sums of pro- 
ducts, of digits perpendicular to each other in each 
of 5 positions, as follows: 


Position : 5 4. 3 2 1 
Nature of Products. 1o-thousands. thousands. hundreds. tens. units. 
321 321 321 A Ee WA | 
123 123 123 123 123 


ee ee 


mews arc (10.4 


36 LIGHTNING CALCULATORS. 


61. Arithmeticians generally are aware of the 
existence of this method; but comparatively few 
conquer the first discouragements attending its 
practice. Lightning calculators, like poets, are 
born, not made; and their exceptional powers can- 
not be transferred to their less gifted admirers. 
With the operator of only average capacity, this 
method increases the lability to error, and dimin- 
ishes the facility of its detection. 

‘¢And yet shew I unto you a more excellent 
way.’—I Cor. XII, 31. 


THE GEOMETRICAL BIRD. 37 


THE GEOMETRICAL BIRD. 


ee 


62. In the miniature Multiplication Table (19), 
the product of 1212 is developed by adding the 
four numbers in a certain direction. 

The accompanying diagram instead of vias 
consisting of four squares of uniform boy 4 
size, represents the space occupied by 

12 rows of 12 buttons each, 10 2 
separated by crossed threads 
into four sections, correspond- 
ing to arithmetical syllables. 
The relative proportion of these 


syllables suggests the similitude 
of four parts of a bird, viz: 


1 ten-ten, the body, 100 buttons. 
2 tens, a wing, PAB ied 
2 tens, a wing, open 
2 twos, the head, Ao 


63. Dispensing with the formal lines of a table, 
a ready method is here suggested of obtaining the 
products of factors from 11 to 19, by either a 
written or a mental process, as shown at the top of 
next page: 


38 THE GEOMETRICAL BIRD. 


Written process: Mental process: 
1 
EEX 13S salero 11+3 tens, 143 
3 1x3 units 
143 
2 
ISSAG AAG ae 13+6 tens, 208 
6 3x6 units, 
208 
7 
1 Xa 17+7 tens, 989 
ff 7X7 units, 
289 
5 
15x19 1456 15+9 tens, 28h 
9 5x9 units, ; 
285 


64, The same method may be applied to larger — 
factors, but not with any practical advantage, thus: 


4 
4 4 
4 4 3) 4 
24 34 3 44 
24X27 28 34x37 328 34x47 3h or 4.28 
BY 37 3 47 
7 7 7 
er 7 7 
648 


1258 1598 


THE GEOMETRICAL BIRD. 39 


65. <A better practical solution of the last prob- 
lem—by more multiplication and less addition— 
consists in arranging the partial products in dia- 
mond form, thus: | 


eer 

ao 4 
4x4 tens, wing, 16 
4x3 ten-tens, body, 7x4 ones, head, 1228 
7X3 tens, wing, oy) 


or the 4 syllables may be condensed to to 2 double 
syllables, thus: 


4x3 and 7x4 body and head, 1228 
4x 4+-7 3 tens, wings, 370 


By either method the full product may be read 
from left to right. The last process should be 
purely mental, written figures being entirely un- 
necessary except for beginners, or for persons very 
deficient in powers of computation. 


66. In like manner should be solved—without 
written figures—such problems as the following: 


Ha, (86.97. San 6 
Suny, 
8x9 and 6x7, body and head, 724 2 


9) 
8x7+6 x9 tens, wings, 110 (0) 8342 


40 THE GEOMETRICAL BIRD. 


Ex. 64x79. 
6x7 and 4x9, 4236 5056 
6x9+4x7 tens, 820 


The old rule, “ Write the multipler under the 
multiplicand, units under units, tens under tens,” 
etc., need not be observed. The curved lines in 
the following example may serve to fix in the 
memory the details of this horizontal process—ex- 
pertness in which will follow a small amount of 


practice: 


LU hy ab-tolo dy ah: 


a 


28x73 2x7 and 8x3, body and head, 1424 
2044 


eek See 


28x73 2x3+8x7 tens, wings, 620 


eee 4 


67. When the head product is expressed by 1 
digit, the place of tens must be filled by a cipher. 


Bx 32 53. 
3x5 and 2x3, 1506 1696 
3x 3+2x5 tens, 190 


Ex. 42 54. 


4x5 and 2x4, 2008 29 
4x4+425, tens, 260 


THE GEOMETRICAL BIRD. 41 


68. In the preceding examples, the wing pro- 
ducts have necessarily been computed separately 
and then added; but in the following examples a 
digit common to both factors serves as a multiplier 
of the swm of the other two digits. 


Ex. 34x 84. 
3x8 and 4x4. 2416 
4X11 tens, 440 ae 
Ex. 65 x 68. 
6x6 and 5x8, 3640) 445 
6X13 tens, 780 
Ex. 4648. 
4x4 and 6x8. 1648 
414 tens, 560 rate 


69. In square numbers, the head and body are 
both square, and the wings are alike. In compu- 
ting wing products, therefore, either digit may serve 
as the multiplier of the other one doubled. 


Ex. 742 

72 and 42 4916 

7X8 (or 4X14) tens, 560 5476 
Ex. 89? 

82 and 92 648] 

8x18 tens, 1440 7921 
Ex. 982 

92 and 82 8164 | ogo4 


916 tens, 1440 | 


42 ; THE GEOMETRICAL BIRD. 


70. Operations thus far have been confined to 
the actual partial products of the factors employed. 
Rapid and easy as the method may seem compared 
with the old-time ciphering, it is to be followed by 
far quicker and easier forms—perhaps less trans- 
parent occasionally—under the heads of Variations 
of Spelling, Relative Factors, Equivalent Factors, 
Aliquot Division, etc., by which the product of diffi- 
cult factors is deduced from that of easier ones by 
computing the difference between the two. 


, 71. In any case of doubt as to correctness of 
result, caused by imperfect comprehension of the 
method on the part of the student, he should be 
able to fall back instantly on the ground-work 
already established,—-with which he should make 
himself thoroughly familiar before attempting to 
study further. To this end he may with great ad- 
vantage prepare a number of blank tables, (36), in 
which to write the squares of numbers from 11 to 
99 as he shall mentally compute them. A record 
of the time spent on each table, and of the number 
of errors committed may be made an index of his 
progress in rapidity and correctness. 


THE DOUBLE-HEADED BIRD. 43. 


THE DOUBLE-HEADED BIRD, 
Or, Variations in Spelling. 


72. Variations in the spelling and syllabication: 
of words are of frequent occurrence, as Plow, Plough,. 
An’-ti-podes, An-tip’-o-des. In the Roman notation,,. 


4 is expressed by IIIT or IV. 
9 VIIII or IX. 
40 % XXXX or XL. 


So the number 27, for example, may be con- 
sidered as 20+-7, or as 30-—3; and its square may be 
deduced from that of 30 by subtraction, as readily 
as from that of 20 by addition. 900 buttons. 
arranged in rows as the square of 30 may be rednced 
to the square of 27 by removing 3 rows from a side 
and 3from anend. ‘The first remove is 3x30, the 
second 3X27; but this clumsy computation may be 
avoided by primarily conjoining the square of 3 
with that of 30, pre- 27 3 
cisely as in compu- 
ting the square of 33, 
thus, 3030 and 3 
x3=—909. This dou- 
ble syllable is here 
represented in dia- 
gram. Theaddition — 


of 2 external wings, 


44 THE DOUBLE-HEADED BIRD. 


each 3X30 would complete the square of 33; 
while the subtraction of 2 internal wings, repre- 
sented by the dotted lines, each 3X27, and 2 
heads, each 3X3, or 6X30 in all, would develop 
the square of 27; thus: 


332 
30 x 30 and 3x3, 909 
6X30, 180 ee 
272 
30 x 30 and 33, 909 ) 
6X30, 18G(co 


The only difference in the two operations is the 
final subtraction, instead of addition of the second 
syllable.* 


73. This method of computation may be con- 
veniently applied to numbers ending with 6, 7, 8, or 
9; as per examples on next page. 


* The difference between the squares of any two numbers is 
equal to the sum of the numbers multiplied by their difference; 
thus, between 


272 and 302 67x3=171, fis0—9]. 
302 and 332 63x3=189, {180+9], 
272 and 332 60x6=360. 


612 and 692 130x8=1040. 


RELATIVE FACTORS. 45 


462 502 and 42 2516--8 x 50= 2116 
672 702 and 3? 4909-—6 x 70= 4489 
782 802 and 2? 6404—4 x 80 = 6084 
892 902 and 1? 8101—2 x 90=7921 
36 x 39 402 and 4x 1 1604—5 x 40 = 1404 
46x 48 502 and 4x 2 2508—6 X 50 = 2208 
66 X 67 702 and 4x3 4912—7 x 70=4422 
87 X98 10C C and 13x2 10026—15 C =—8&526 
73X89 100 C and 27X11 10297--38 C =6497 


oes ee a SeSeSe: 


RELATIVE FACTORS. 


74. There is a curious relationship in numbers, 
heretofore unrecognized in arithmetical text-books, 
whereby a product difficult of computation, as that 
of 8273, is deduced from one easily computed as 
that of 80x 75,—the substitute product, the differ- 
ence between the two, and the question whether 
that difference is plus or minus, all being instantly 
apprehended. The term RELATIVE Factors has been 
chosen to designate in this work such interchange- 
able numbers 


75. Relative factors are factors of an equal 
sum. In this example the sum of each set is 155. 


46 THE FORTY-INCH BAND. 


Any set of factors may be resolved into a relative 
set by adding to one factor and subtracting from 
the other, any chosen number; the object being to 
secure a number easily computable, usually a mul- 
tiple of 10, as one of the new factors. In this 
example the number-2 has been subtracted from 82, 
and added to 73. 


— 


THE FORTY-INCH BAND. 


76. One hundred 1l-inch cubes, 10*10=100 
arranged as the square of 10, may be 9x11= 99 
inclosed by an endless band of rib- 8 X12= 96 
bon or tape 40 inches long. But10 7x13= 91 
such bands may be symmetrically 6x14= 84 
and compactly filled by 715 cubes, 5X15= 75 
arranged in 10 geometrical fieures, 4x16= 64 
no two of them alike in number or 3x17= 51 
form, as indicated in theaccompany- 2x18= 36 
ing table. lx 19S 


77. Here are 10 sets of relative factors, which 
notwithstanding the wide variation of their products, 
have yet a common sum, 20, equal to 2 adjacent 
sides of each figure, however unequal those sides 


RELATIVE FACTORS. 47 


may be. The last product compared with the first 
is minus 9 tens in one direction and plus 9 ones in 
another, but the 40-inch band fits as tightly around 
the 19 blocks as around the 100. 


78. The product of the equal factors 10x 10 is 
the square number 100. The other nine products 
correspond with the differences between 10? and the 


nine lesser squares, as appears by subtracting from 
100 the 


Sqnare numbers, 1l 4 9 16 25 36 49 64 81 
Leaving as rom’drs,99 96 91 84 75 64 51 36 19 


79. Relative factors are distinguishable as inner 
and outer, the inner being equal and the outer 
unequal; or, if the inner are unequal, the outer are 
are more so. For example, 


Inner, 10x 10 9x11 80x75 
Outer, Bee ele 8 se LD Soe ee 

The product of the inner factors is invariably 
ereater than that of the outer. 


80. The difference in product is equal to the 
product of the differences; or, the product of the 
outer factors plus the product of the differences is 
equal to the product of the inner factors. The 


48 RELATIVE FACTORS. 


differences are taken between two factors of one set 
and one factor of the other,—usually the selected 


decimal number. 


81. These principles are applied to the problem 
8273, as follows: 


Original factors, 89 anne 

Relative factors, (inner, ) 80x75 = 6000 

Differences, (from 80) 2, ea 14 
Required product, 5986 


The number 80, 2 less than 82, being selected 
as one relative factor, the other must be 75, 2 more 
than 73. The new factors being inner, their pro- 
duct is too large, and must be diminished by the 
product of the differences. The differences between 
the chosen factor 80 and the original factors 82 and 
73 are 2 and 7, and their product is deducted from 


that of the relative factors. 


82. The same result may be obtained from 


outer factors, thus: 


Original factors, 73 X 82 
Relative factors, (outer, ) 70... 85 (=eoeae 
Differences, (from 70) 3x12 36 


Required product, 5986 


RELATIVE FACTORS. 49 


83 Several examples of this method of compu- 
tation, as applied to factors from 11 to 19 have 
already appeared (63), the number 10 being in 
each case one of the relative factors. 


84. The difference in product of 2 sets of rela- 
tive factors is independently computable. Thus the 
unknown product of 58 x 63 
exceeds that of Dim ere + 
by the product of differences (from 58) 1 x  6=6 


This may be demonstrated by 58x63 buttons. 
If 1 row be taken from the side and 1 added to the 
end, the figure becomes 57X64: but the side row, 
63, exceeds by 6 the number required for the end 
row, 57. So with 75x80 buttons changed to 
73X82; the 2 side rows, 80 each, exceed by 2X7 
the number required for 2 end rows of 73 each. 


85. In the practical application of this method 
mental confusion may be avoided by observing the 
following order: 


1. Computing the product of differences be- 
tween the original factors and the chosen relative— 
leaving the second relative momentarily out of 
consideration. 


50 THE BROAD-SHOULDERED BIRD. 


2. Distinguishing this product as plus or minus, 
to or from. 

3. Computing the relative product. 

4. Adding or subtracting the difference pro- 
duct. 


Examples: 
57 x 64 3x4 from 60 61=3648 
58x63 23 from 60 61=3654 
57 X64 7X14 to 50X%71=3648 
58X63 8x13 to 50X71=3654 
48x 46 2x 4to 50x 44—2208 


THE BROAD-SHOULDERED BIRD, 
and the Headless Bird. 


86. The accompanying diagrams demonstrate 
the soundness of the foregoing method of deducing 
inner products from outer factors, or outer products 
from inner factors; the first, by the broad-shouldered 
bird, the second, by the headless bird. 

I. 24x24, 
4x4 to 20x 28=576. 


~ 


THE BROAD-SHOULDERED BIRD. 51 


The first syllable in this computation is the 
head, the second the body and 2 wings. Fig. 1 
represents the space of 24 rows of 24 buttons each, 
separated by crossed threads into 


20 x 20 
4x20}and 4x4. 
4x 20 
Bigwl, Fig. 2. 
4 
4 4 be] 
20 20 


| 
| | 


In Fig. 2 the upper wing is removed to the side, 
and the head to the center, the square bird thus 
becoming broad-shouldered. 


Ex. 34 x 37. 4x7 to 30 x 41=1258 
34 X 37 6x3 to 40 X 31=1258 
46 x 48 6x8 to 40 x 54=2208 
IT. 16x24. 


42 from 202—=384 


52 THE BROAD-SHOULDERED BIRD. 


Fig. 1. | Fig. 2. 

eres ) 4 | 
ae a i 
ue 


4 16 + 16 4 


. Fig. 1 represents the space occcupied by 24 rows 
of 16 buttons each, separated by parallel threads 
indicating the body, 16 rows, and 2 wings, 4 rows 
each, of the headless bird. In Fig. 2, one of the 
wings is removed to the top of the body, forming 
the square bird, 20 x 20, minus the head, 4x 4. 


Examples: 
34 x 46 62 from 402=1564 
34x48 72 from 412= 1632 
66 x 94 142 from 8026204 


87. The student may now extend the table of 
square numbers to 109? or 1192, thus: 


1092 92 to 118C=11881 
1112 112 to 122C=12322 
1192 192 to 1388C=14161 


88. When the sum of the units of the factors 
is equal to 10, both relative factors may be decimal 
numbers, as will be seen on the next page. 


THE BROAD-SHOULDERED BIRD. 53 _ 


22 x 28 2x8 to 20x30= 616 
22 x 38 2x18 to 20x 40= 836 
34 x 56 4x26 to 30 x 60=1904 
76 x94 6x24to 700 =7144 
43 x87 3x47 to 40x 90=—3741 


89. Any number ending with 5 may be squared 
by annexing 25 to the product of the nearest tens, 


as 
752 25to 7X 8= 5625 
352 25 to 3X 4= 1225 


1952 25 to 19 X 20=38025 


90. From the squares of numbers ending with 
5 may be deduced the products of other numbers, 


as, 
68 x 82 72 from 75%= 5576 
BT 43 82 from 352= 1161 
104 x 126 112 from 115213104 


131 x 139 | 42 from 1352= 18209 


54 EQUIVALENT FACTORS. 


EQUIVALENT FACTORS, 


91. Factors of like product are termed Equiva- 
lent Factors. If one factor be multiplied and the 
other divided by any given number, equivalent 
factors are developed, as 


Factors, 12*%a2 
Equivalent factors, + and + of 12 3x48 
1 and 3 4x 36 pee 
| and 2 6 x24 {aia 
2 and 3 8x18 my 
3 and 4 9x F 


92. Fractional factors, or factors widely un- 
equal, may often be advantageously resolved into 
their equivalents, and these, perhaps, into relatives, 
as in the following examples: 


Factors. Equivalents. Operation. Product. 
24 x96 482 22 to 50x 46= 2304 
13 <x 93 B9pea1 9 to 30x40= 1209 
274 x 616 110x164 1640 to 164 C=18040 
47 x194 94x 97 4x7 to 91C= 9128 
29 x84 58+ 42 82 from 502= 2436 


174 x 184 69x46 4x19 from 65x 50= 3174 © 


EQUIVALENT FACTORS. 55 


93. The equivalent factors of 202, 502 and 100, 
serve as convenient bases for the squaring of many 
other numbers. The factors, 20x20, one being 
divided and the other multiplied by 2, are resolved 
into their equivalent factors, 10x40. Any number 
from 11 to 34 may be readily squared by subtract- 
ing 10 therefrom, multiplying the remainder by 40, 
and to the product adding the square of the differ- 
ence from 20; thus: 

912 «11x40+ P= 441 
27» 17x40+ To= 729 
342 24 x 40+ 142=1156- 


192 9x40+ 12=361 
162 6x40+ 42=256 
122 2x40+ 82=144 


94. This method may be demonstrated as fol- 
lows: —If 400 buttons be arranged as the square of 
20, the square of 21 may be produced by adding 40 
as 2 wings and 1 as the head, or the square of 22 by 
adding 2x 40 as wings and 2? as the head. Now if 
the original 20 x20 be considered as 10x40, the 
square of 21 is perceived to be 11x40+1, that of . 
22, 12x 40+ 22, etc. 


95. The factors 50 x50 may be represented by 
their equivalents, 25100. Any number from 36 


56 EQUIVALENT FACTORS. 


to 64 may be squared by deducting 25, and to the 
remainder considered as hundreds, adding the 
square of the difference from 50, thus: 


512 260+ 12=2601 
522 (270+ 222704 
642 390+ 1424096 
472 290+ 322209 
372 12C+132=1369 


96. This method may be modified as follows: 


To the units of numbers 


above 60 add 35, 
above 50 add 25, 
above 40 add 15, 
above 30 add 5, 


and to the sum taken as hundreds, add the square 
of the difference from 50, thus: 


382 8+ 5—130+12,=1444 
432 3+15=18C+ 721849 
562 6+25—31C0+ 62=3136 
612 1+35—360+4+ 1lo—3721 


97. The factors 100 x 100, one being halved and 
the other being doubled, are readily resolved into 


EQUIVALENT FACTORS. 57 


their equivalents, 50100. Any number proxi- 
mate to 100 may be squared as follows: 


1. By deducting 50 therefrom, multiplying the 
remainder by 200, and to the product considered as 
hundreds adding the square of the difference from 


100. " 
CHx, 992 49x 200+ 1o= 9801 


88? 38 X 200+122= 7744 
113? 63 X 200 + 132= 12769 


2. By doubling the number, diminishing the 
product by 100, and to the remainder, as hundreds, 
adding the square of the difference from 100. 

Ex. 982 1 96—100= 960+ 22—= 9604 
Bo (1L)78 Saou its £994 
1122 2 64-100 =124C + 19?— 12544 

98. By amore practical process the difference 
Jvom 100 of the number to be squared is multiphed 
by 200, the product added to, or subtracted from, 
100C, and the square of the difference added, thus: 

1062 6 x 200+ 100C + 62 11236 
932 7 X 200 from 100C +7? 8649 

99. With unlike factors, the sum of the differ- 
ences, in hundreds, is added to or subtracted from 
100C and the product of the differences added, thus: 


107 x 122 29C to 100C + 7x22=13054 
88 x 94 18C from 100C +12x 6= 8272 


58 MULTIPLICATION BY ALIQUOT DIVISION. 


MULTIPLICATION BY ALIQUOT DIVISION. 


100. Aliquot Division is often made a practical 
substitute for multiplication. The aliquot parts of 
100, corresponding to the divisions of Federal 
money, as the half-dollar, the quarter, the “ Yankee- 
shilling,” 3, the “York-shilling,” 4, with various 
other fractions, are so familiar that we are hardly 
conscious of a mental effort in computing by them. 
For example, the value of 44 yards of calico @ 123 
cents is rarely treated as a problem in multiplica- 
tion. To mathematical man and unmathematical 


woman alike, it seems almost a matter of instinct 
to divide 44 by 8. 


101. The practice of this method has been 
usually confined to such familiar numbers as repre- 
sent exact and easily computed divisions of 100: 
bnt by a very simple expedient the process may be 
extended to countless proximate numbers by means 
of an easy computation and a difference, that differ- 
ence being a percentage. 


102. The problem 68 xX 50 for example, by any 
practical arithmetician is instantly perceived to be 


MULTIPLICATION BY ALIQUOT DIVISION. 59 


6800+2=3400, while the problem 68x48 or 
6800-21, seems to be practicably incomputable by 
mental process. But as 


2x 50 equals 100, and 
2x48 equals 96 


it appears that 50 less 4 per cent. equals 48; and it 
follows that there must be a corresponding differ- 
ence in any products of a common factor by these 
two numbers. Then 


48 x 68=50 x 68 (3400) less 49% (136)=3264 
or 48 x 68=4800 x 2 (3200 plus 29% ( 64)=3264 


The second process is based on the proximity of 68 
to = of 100, it being 2% in excess, as thus appears, 
68 + 34=102. 


103. The accompanying table contains a limited 
collection of aliquot factors, together with many 
proximate numbers, and the percentage of their 
variation. The fractions in the upper line corre- 
spond with the numbers beneath them as aliquot 
parts either of 100 or of numbers varying there- 
from by the percentage expressed in the column at 
the right. Other aliquot factors may be frequently 
utilized, as 15 x 62%, 20x 5, 30 x 34, 80 x 14, ete. 


60 MULTIPLICATION BY ALIQUOT DIVISION. 


TABLE OF ALIQUOT AND PROXIMO-ALIQUOT 
FACTORS, 


] 2 3 Variation 


A 2 1 
3 2 3 4 : from roo, 


30 | 45 | 60 | 672} 90 |-109% 


104. Examples of multiplication by aliquot 


division will be found at top of next page. 


RECAPITULATION. 61 


48 x 46 14600 2300—40/=2208 
67 X93 2 x 9300 6200+ 40 = 6231 
43 x78 3x 4300 3225 +406 = 3354 
84+172 4 x 8400 1400+5%=1470 
33 x 87 4 x 8700 2900—19% = 2871 
23 x 87 1 8700 2175—8% =2001 
32 x 86 1 x 8600 2867—40% = 2752 


In the last example the proximate numbers 
2867—115 are substituted for the exact numbers, 
28662—1142=2752. 


---— 


RECAPITULATION. 


105. The various methods of computation set 
forth in the preceding pages may be briefly recalled 
by their application to a single problem, reference 
being made to the appropriate section for the devel- 


opment of each process. 


Example 46 x 48. 


Section. Kasy Computation. Difference. 
68 40x40 and 6x8 +14x 40 
86 40 x 54 TAKS 
85 50 x 44 msec ae 
73 50 x 50 and 4 x 2 — 6x50 
90 47 X47 — 1 
99 46 x 50 --46X 2 


104 4600 x4 ++ 40¢ 


62 RECAPITULATION. 


106. Noone of these methods seems entitled 
to universal pre-eminence. The first, which seems 
the most difficult and undesirable for this particu- 
lar problem, may seem the easiest and best, applied 
to another one, as 42x44; and thns each method 
in turn may have preference, according to the char- 
acter of the factors. The ability to determine at 
sight the most feasible method of treating 2 fac- 
tors, must depend largely upon a quick perception, 
and a discriminating judgment, both sharpened 
by persistent practice. 


107. ‘Tables like the one below, presenting in 
small space a great variety of exercises, may be 
used with much benefit, the factors being frequently 
changed, and the mode of computation being con- 


stantly varied, as may be seen most advantageous. 


RECAPITULATION. 63 


108. ase, rapidity and correctness in compu- 
tation, whether considered as a natural endowment, 
a hard-earned acquirement, or a happy combination 
of both must ever vary largely in different indi- 
viduals. Those who fail in these mental exercises, 
may yet find profit in the written system which is 
to follow; while those who have succeeded hitherto, 
may advance to still higher proficiency. 


64 WRITTEN MULTIPLICATION. 


WRITTEN MULTIPLICATION. 


109. Inpassing from 2 two-digit factors, with 
4 partial products, to 2 three-digit factors, with 9 
partial products, the bounds of mental computation, 
for the average operator, are transcended; and 


assistance must be had from a written process. 


110. There is an intervening border-land, occu- 
pied by multiplicands of 3 or 4 digits with multi- 
pliers of 1 or 2, wherein mental or written methods 
may have precedence according to the proficiency 


of the operator. 


111. Products of such factors may be compared 
to long words, in contrast to the square words, 
resultant from equal factors. Thus the product of 
9x 9999, and that of 99x99 are each made up of 
the 4 syllables 81, with a variation in their relative 


position, as here appears: 


81 
ox 9999 8161 09 X00 8161 
8181 81 


89991 9801 


WRITTEN MULTIPLICATION. 65 


In square feet, one represents a modest garden- 
plat; the other, a lane 9 feet wide and nearly 2 
miles long. 


112. In the multiplication of 2 digits by 1 digit, 
the 2 syllables should be instantaneously appre- 
hended and combined, reading from left to right by 
a double thought, thus: 
3X 83. 
3x80 and 3x3 249 
3X 84. 


3x80 and 3x4 24 
12 252 


In the first example, the syllable 9, expressed by 
1 digit fills the place of the cipher otherwise re- 
quired for the full expresion of the tens syllable, 
240. In the second example, the units syllable, 12, 
overlaps the tens syllable 24, and*the tens must be 
combined by addition, two hundred-forty-twelve thus 
becoming 252. In many cases of this kind it may 
be advantageous to think in tens, thus: 24+1.2= 
25.2 tens or 252. 


113. The product of 2 digits may frequently be 
taken as 1 syllable and thus the range of mental 
computation be extended to factors of 3 or more 
digits, as will be found on the following page. 


66 WRITTEN MULTIPLICATION. 


9X 128 (12-8) 108 tens. 

7.2 115.2 
8X713 (7-13) 56 

104 5704 
6 x 1216 (12-16) 7296 
4x 6182 (6-18-2) 24728 


114. The product of digits separated by ciphers 
may be apprehended at sight, as 


8 x 604 4832 
12x 7002 84024 
14 60703 849842 


In these examples the syllables stand side by 
side without overlapping, but in the examples follow- 
ing, digits being substituted for ciphers inthe multi- 
plicands, the syllables of the Ist, 3rd and 5th 
digits are taken in one line, and those of the 
2nd and 4th in another. 


8 X 634 8x 604 4832 
8x 30 240 5072 

12x 7482 12x 402 4824 
12x 7080 8496 89784 

14x 62743 14x 60703 849842 


14x 2040 2856 878402 

115. In adding these partial products, instead 

of beginning at the right and carrying tens to the 
left, it is recommended that the operator read com- 
prehensively from the left, without writing the pro- 
duct, iucreasing the footing of each column by the 


tens anticipated from the next one. 


THE DIAMOND FORM. 67 


THE DIAMOND FORM. 


116. In multiplying 3 digits by 3 digits, the 
diamond form of arranging the 9 syllables has been 
fore-shadowed in an improvised table (29). Dispens- 
ing with the formal lines of a table, this method of 
arranging partial products, or its modifications, may 
be advantageously adopted in practice; being prefer- 
able to the old process in the following particulars: 


1. As there is nothing to carry countless errors 


are avoided. 
/ 


2. The partial products being computed, or 
rather recollected, faster than they can be written. 


The brain rests while the hand works, 
a grateful relief from the old order of things, in 
which, owing to the complications incident to 
carrying, . 

The brain works while the hand waits. 


117: 1n practicing this method the factor figures 
are placed sufficiently apart to allow twice their num- 
ber of figures in line beneath them. Lach partial 


68 THE DIAMOND FORM. 


product falls mechanically into its place, either 
squarely under its factors when they are of the same 
order, or under a point mid-way between them 
when they are of different orders. In this example, 
684 x 397, the partial factors are expressed in full 
instead of by their initial digits alone. , 


684 
B67 

6007 49 
600% 90, 80x? 5456 
600 x 300, 80x90, 4x7 187228 
8x 300, 4x9 2436 

4x 300 | 12 
271548 


118. The square of 444 is here computed in a 
similar manner, the factor being written only once. 


444 
400 x 4 16 
400 40, 40K4 1616 
4002 402 42 161616 
40x 400, 4x 40 1616 
4x 400 16 


197136 


THE DIAMOND FORM. 69 


ILLLUSTRATION. 
c 400%4 th aacg 
5 400 x 40 sox0||s 
< c 
S 
— 
=H 
=u 
> ae 
es 400 x 400 NM 
> 
= 
400 40 4 


‘¢ Hach one had six wings; with twain he covered 
his face, and with twain he covered his feet, and 
with twain he did fly.” —Isa. VI, 2-3. 


Diagrams on the same scale representing the squares 
of 4444 and of 44444 would measure on each side, 
the first 22 inches. the second, 18 feet. 


119. Product syllables arrranged for addition 
must each be represented by 2 figures, a cipher 


70 THE PYRAMID FORM. 


being prefixed when the syllable is less than 10, 
except that at the extreme left the cipher may be 


omitted. 
Example 31632. a Es 
3X3 09 
SC Os ulin oO 1803 
DS OKy LO, Oe 020618 
32 12 62 32 9013609 
1X3, 6561)3 506 030618 
6x3, 3X1 1803 
3X3 09 
10004569 


120.. The number of figures in any product is 
either equal to the number in both factors, asin the 
squares of 32, 317 and 3163, or one figure less, as 
in the squares of 31, 316 and 3162. 


THE PYRAMID FORM. 


121. The diamond form may be condensed, and 
the partial products arranged in pyramid form, by 
coupling syllables of like order. 


THE PYRAMID FORM. Le 


Example 684 x 397. 6 8 4 
397 

6xX7+3x4 54 
6x9+3x8, 8x7+9x4 7892 
6x3 8x9 4X7 187228 
271548 


Coupled syllables may be written separately, 
except at the extreme left, when their sum exceeds 
99, as in the following example: 


Ex. 897X789. 789 
897 

7X7+8x9 121 
7X9+8Xx 8534 12739 
meres; 9X7 567263 
707733 


122. In squaring a number it should be remem- 
bered that one factor must be doubled in computing 
wing products. 


Ex. 438. 4 38 
8x8 64 

3x3, 6X8 2448 

42 32 So 160964 


191844 


THE INVERTED PYRAMID FORM. 


THE INVERTED PYRAMID FORM. 


123. Practically, this form is preferable to the 
preceding ones, as the products of digits in perpen- 
dicular lines are taken first, and thereby the posi- 
tions of the several orders of units are determined 
at the outset. 


Ex. I 721834 Yaa a 

8 3 4 

7x8 2x3 Tose 4 560604 

7X38 2) 94 35] 3711 
7x4+8x1 36 

601314 

Ex. Il. 4372 8694. 8694 

; 4372 

8x4 6x3 9x7 4X2 32186308 
8x 3+4x6, 6xX713x4, 9Xx217x4 486946 
8x714x9, 6X2+43x«4 92294 
8x2+4x«4 32 

38010168 


124. As each digit in a factor is represented 
by 2 figures in its product syllable, so each cipher 
must be represented by 2 ciphers. 


THE INVERTED PYRAMID FORM. 73 


Hx. I. 2703x8976. 2703 
Ex. IT. 3024 2850. S-¥eieG 
302 4 . 12 
28 5 0 1442 

— 184900 
06001000 16630018 
241620 560021 
1932 0027 
08 24 
8618400 24262128 


125. In examples lke the following, the entire 
product may be written in one line, or may be com- 


puted mentally. 


Ex. 307 x 906 7009 x 4008 
307 i 0;,.059 
9 0 6 4008 
278142 28092072 


126. With factors of an unequal number of 


digits the product forms are incomplete. 


Ex. 6789 x 24. 6789 
24 
6789 24 
24 1228 
$s * 1432 
01636 * 1636 
* 1450 ke 
1228 es 
24 * 


162936 162936 


74 RAPID MULTIPLICATION. 


RAPID MULTIPLICATION 


127. Expert mental computers may abridge 
the written formula, and perform more rapid work 
by dividing the factors partially or wholly into sec- 
tions or periods of 2 figures each, thus making the 
product syllables larger in amount and less in 
number. The product of a period of 2 figures must 
occupy 4 places, ciphers being prefixed if occasion 


requires. 

Ex. 1312x914. Lge 
ooied 
139, 12X14 1170168 

13x 14+12x9 (”) 290 
1199168 
269 X 248. 269 
248 
26x24, 9x8 62472 
25x17—1 ~~ (°) 424 
66712 
45672, 4567 
452 672 20254489 

90X67 (°°) §030 


20857489 


MULTIPLICATION. BY SUBTRACTION. 75 


7128x7246. peed) ities 
7-9 486 
“x7, 12x24, 8x6, 49028848 
area (2) 12% 22 (°) 952264 

Tx 14-(°°) 98 
51649488 


MULTIPLICATION BY SUBTRACTION. 


128. The product of any number multiplied by 
nines may be determined by annexing a cipher for 
each nine and then subtracting the original 


number. 
Ex. I. 76499. 76400 
764 
75636 


DemonstTraTIon. 764 ones from 764 hundreds 


leave 764 ninety-nines. 


ieee e900 C099, 


9999000 or 9990000 
9999 wou 


9989001 9989001 


76 MULTIPLICATION BY SUBTRACTION. 


129. With factors of a like number of figures, 
the process—either written or mental—may be 
abridged by annexing to a number J less than the 
multiplicand the successive remainders obtained by 
subtracting each figure of that number from nine, 
commencing at the left. 


Examples: | 
68x99 6732 
695 999 694805 
7890 x 9999 78892110 
7891x9999 78902109 


The first example is thus performed: 


67 with (6 from 9) 3 and (7 from 9) 2 annexed. 


130. Demonstration. 68 rows of 99 buttons each, 
changed by taking one row from the side and adding 
one to the end became 67 rows of 100 each. The 
side row, 99, being 32 in excess of the end row, 67, 


the total number must be 6732. 


Again, the inner factors, 68 x 99, 
exceed the outer factors, 67: °>e aie 
by the product of differences (from 68)l1 xX 32. 


See Relative Factors. 


BREVITY. ak 


lA ae 


131. Philological “Josh Billings” playfully pities 
punctilious pundits, who with infinite labor acquire 
just education enough to spell a word in one partic- 
ular way; but not enough to indulge themselves in 


66 


that variety of forms wherewith he, “ good easy 
man,’ may give his word a new dress—usually a 


very short one—for every day in the week. 


132. The adept in the various short and easy 
methods developed in this work, may well be moved 
to a more hearty compassion for the unprogressive 
cipherer who knows only one process of multiplica- 
tion, and that a laborious, blind, and often inaccur- 
ate one. Whether ‘‘fonetik orthografy’”’ shall ever 
overcome popular opposition or not, no reasonable 
objection can be made to shorter, surer, and more 
intelligble methods of spelling products. The 
brevity of free and easy spelling is illustrated in the 
treatment of the following 10 problems, each of 
which is solved in 2 or more different ways. The . 
operations extend only to the development of partial 
products from which the full products may usually 
be read from left to right. 


78 BREVITY. 


Problem. First Process. Second Process. 
l. 1962. 36186 40016 
228 1s 
LR TO OE: 17412 1) 218 
526 40% 872 
3. 825x426. 320650 (8) 3550 
308 (—19%) 3550 
4. 9864297. (25) C50). pn 
| at x28} 76 gov 5728 
ee x64 1728 
De SLIT Sas 13606 « 103 17716 
67 —>x 20 344 
6) 1400 aoe 13524 x 40 18120 
| 369 5% . 906 
26428 X77); 4482100 x750 4821000 
4996 < 200eteu 
8. 46.52 1642.25 1 2325 a. 
520 =e 162.75 
9. . 32502. 9062500 4 10833333* B. 
150 — le ee 
1GS6 C87 Be 121645625 fc 1484375 9c 
176 —59% 7421875 
1200 : 
1650 
} Third Process. 
A. 46X47 and .25 2162.25 
B. 3233 and 2500 10562500 
Cc. 118-+19.5 and 752 137505625 
18 19.5 351 


* This product must be expressed by 8 figures, the last 4 being 
2500 (119, 36.) 


ERRORS. 79 


ERRORS: 


133. The way to avoid errors in computation is 

_to avoid them.* There is no other way. ‘The 
"process of multiplication being learned in childhood, 
if in after life all its conditions be strictly observed, 
error is impossible, and each and every attempted pro- 
duct must infallibly be right. Failing this, the 
defect is not in knowledge nor in memory, but in 
attention and precision. When Grandame ‘‘ drops 
a stitch,’ it is not because knitting is too intricate 
or too laborious. Nothing can be easier than for 
her to make a correct stitch; but in the monotonous 
making of a thousand, she becomes tired or sleepy 
or inattentive. Her boys and girls all inherit in a 
greater or lesser degree, this trick of inattention; 
and consequently as errors in computation are 
likely to continue for some time to come, methods 


must be practiced for their detection and correction. 


_ * Horace Greely said: “The way to resume specie payments, 
is to resume.” 


80 PROOFS WHICH DO NOT PROVE. 


PROOFS WHICH DO NOT PROVE, 


134. Under the fallacious title of PROOFS, some 
arithmeticians have prescribed certain forms, which 
like the forms of criminal jurisprudence, sometimes 
result ina conviction of the wrong, never in a vindi- 
cation of the right. When a prisoner is not proven 
to be guilty, his innocence may be legally assumed, 
although that also is unproven, and so when acom- 
putation is not convicted of error, the evidence of 


its correctness is only negative. 


135. These methods of ‘‘ proof’’ are 


30. 
1. Inverting the factors. 

2. Dividing the product by a factor. 
3. Casting out the nines. 


By inverting the factors the multiplier in the first 
operation becomes the multiplicand in the second; 
new partial products are introduced, and the agree- 
ment of the two products establishes a preswmption 
of correctness, strong or weak, according to the 
proficiency of the operator. 


136. Dividing the product by one factor gives 
the other factor as a quotient; provided, that both 


PROOFS WHICH DO NOT PROVE. 81 


operations are correct, which is not proven, and 
also provided that if both are incorrect, one error 
counterbalances another. 


137. Ben. Blunderhead tries his hand at these 
two methods of proof, with the following result— 
his ciphering being omitted : 


Examples: Proof. 
78 X 43=3344 43 x 78=3254 
68 x 92—6246 6246+68= 99 


As the two products of the first example do not 
agree, there must be an error either in the compu- 
tation or the proof, or both. He has doubled his work 
to no good purpose, and now his only remedy is to 
review carefully what he has performed carelessly. 
In the second example he has written 14 figures to 
obtain a false product, and 17 more to prove it by a 
false quotient. Ben’s chief recommendation for a 
clerkship is the fact that he can prove, in the last 
half of each day, the computations made in the first 
half. 


82 CASTING OUT THE NINES. 


CASTING OUT THE NINES. 


138. The question, how the correctness of a 
computation could be demonstrated by casting out 
the nines, has been to the partially informed, a 
mystery deep and awful as that of witchcraft, or of 
the casting out of devils. When the demonstration 
proves to be no demonstration, then the mystery 
vanishes. 


139. Hvery ten equals 9 and 1; every hundred, 
11 nines and 1; every thousand, 111 nines and 1. 
Hence the number, 7468 contains 


In 7000, 777 nines and 7, 
In 400, 44 nines and 4, 
In 60, 6 nines and 6, 
In 8, é 8. 


The 4 remainders are duplicates of the original digits, 
and in their sum, 25, are 2 additional nines and a 
final remainder of 7. This final remainder is the 
essential object of search, the number of nines being 


immaterial; and as the digits of any number may 


CASTING OUT THE NINES. 83 


be considered as so many remainders in excess of 
an unknown number of nines, it is easier to cast the 
nines from them than to divide the whole number 
_by 9. The casting consists in dropping 9 at every 
opportunity from the sum of the digits, as in the 
number 7468, 7+4=—2+6—8 x8=7, or taking 8, 
6, and 4 as 2 nines, the number 7 stands alone as a 


remainder. 


140. An operation in multiplication is tested by 
casting the nines from each factor, multiplying the 
remainders together and casting the nines from their 
product. Then casting the nines from the main 
product, the two remainders should be equal. 
Their inequality demonstrates the existence of 
error. Then equality demonstrates—nothing! The 
product may be right: it may be wrong: but 
according to the theory of the test, it is presumably 


correct. 


141. This device serves as a detective for errors 
not measurable by the number 9; but any falsity, 
however glaring, which involves a multiple of 9— 
including all transpositions of figures—may pass 
the test unchallenged, as appears by the example 


shown on the following page. 


84 CASTING OUT THE NINES. 


B56 was. 
423 4 


32122728 
403657 
6038 
60 


36823098 
1 Error, 36822098 
2 Errors, 37822098 
Tranposition, 36832098 
9 instead of 0, 36823998 
0 omitted, 3682398 
Assumed product, 12 


Remainder 3 
6é 


4 


12 Remamder, 3 


Remainder 3, Presumably correct. 


2, Error demonstrated. 
3, Presumably correct. 


If we know that 12 is not the true product, we 


know it by insight. Itis not disproved by casting 
out the nines. Language should not be perverted 
by applying the term proof to such a piece of no- 


evidence. 


DEMONSTRATION. 85 


DEMONSTRATION, 


142. The positive evidence that an assumed 
product is correct, as distinguished from the nega- 
tive evidence that no error is detected, has its primi- 
tive foundations in the actual counting of tan- 
gible objects representing small products. Every 
civilized child sooner or later demonstrates by his 
fingers that 2 fives are ten. The Multiplication 
Table is a systematic collection of small products 
which have been demonstrated in the counting of 
their squares by thousands of children. Memor- 
izing this table in childhood obviates the necessity 
of innumerable demonstrations by actual counting, 
otherwise required in the daily affairs of life. 


143. From the known in the Multiplication 
Table, the child may push his demonstrations out- 
ward to the unknown, thus: If 12 times 9 are 108, 
36 times nine are 3 hundreds and 3 eights; or if 12 
times 12 are 144, 24 times 12 must be 288. The 
demonstrations of the problems 


13 x 16=208 ( 8) 
99 x 99=9801(13) 


86 DEMONSTRATION. 


are brought within the mental grasp of childhood. 
In fact, the Geometrical Bird method, for all num- 
bers under 100, is from first to last to the adult 
mind, if not to the child mind, a self-demonstrating 
method. 


144. We have witnessed the unsuccessful 
wrestling of Ben. Blunderhead with the problem, 
68x92. (137) Tid. Expert solves it thus: 


80% 8012513 
or (0X) 902% 22-) O26 
or 60X 100+ 8x32 


His mind grasps the problem as a whole: the 
operation is self-demonstrating; and he knows 
that error therein is impossible. But Ben. fails to 
comprehend the process, and so to him there is no 
demonstration; yet there is even for him a partial 
demonstration. The product is the sum of the 
accompanying 4 partial products, which his memory 
may draw from the multiplication table. 

Thus far, he may comprehend that the 72 
work is correct beyond a peradventure. So 54/6 
renouncing all ‘‘ ways that are dark, and 12 


9 


tricks that are vain,’ such as mutilating 


partial products, and carrying from one to another, 


DEMONSTRATION. 87 


or inverting the factors, or dividing the product, or 
casting out the nines, if he can now trust himself 
to read the product-word from the product syllables, 
or to add these numbers correctly, his task will be 
accomplished, the only element of uncertainty being 
in this addition. 


145. To larger minds than Ben’s, larger prob- 
lems, as 684 x 397 may not be demonstrable as a 
whole, but the nine partial products (117), or the 
6 partial products (121), may be examined and 
cross-examined one by one from right to left, or from 
top to bottom, and thus verified as absolutely right. 
Then if correctly added, their sum cannot be other 
than the true product. 


146. Notso with the 3 partial products (30), 
obtained by mutilation of syllables and carrying. 
They cannot besoimplicitly trusted. They are right 
if they are not wrong, but begging the question is 
not sound logic. Universal experience shows that 
this method is everywhere and always a fruitful 


source of error. 


MULTIPLICATION. 


88 


CO ri C2 
Ominrand 
DW l= Ors NM OO HH 
DrOlowiNa aA {Yen} 
DWP O1O HNO ww 10 SH eS 
DOI TNO SH <1 69 6 faa 
Drowdrsio sH OD 1IDSH CO RAM 
Dr OnmNwolo H om 1d CO OMm 
DMP OI AND SH SH 10 69 
TD 6 10 41 GR OD SH ID Heo 
Dre Oily iNow cH 1 
Drornc sas 
Ot IANO 


A TRANSPARENT PROBLEM. 


APPENDIX, 89 


APPENDIX. 
The Combination Table. 


A case has been supposed (29), of a person 
unable, from some mental defect, to memorize the 
multiplication table, and yet able by reference 
thereto to obtain and set in order for addition the 
partial products of factors of any magnitude, by 


means of an improvised geometrical table. 


Proceeding on this principle, a combination of 
unit tables may be so arrangedas to indicate by means 
purely mechanical, the partial products of any and 
all factors, leaving the operator only the task of 
adding them. | 


The accompanying table of four sections, 


one representing units, 
two tens, 
one hundreds, 


consists of unit tables so inverted and arranged that 
factor digits, in distinctive type, run on internal 
lines, one horizontal and the other perpendicular, 
in order that any required partial products may be 
inclosed within a hollow square or rectangle. This 
table is sufficient for all factors up to 99. Factors 
of 3, 4, or 5 figures each, would respectively require 
tables of 9, 16 or 25 sections. 


90 THE COMBINATION TABLE. 


THE COMBINATION TABLE. 
99 x 99 


TENS. UNITS. 


81 | 72 | 63 | 54 | 45 | 36 


72 | 64 | 56 | 48 | 40 | 32 


63 | 56 | 49 | 42 | 35 | 28 21 | 28 | 35 | 42 | 49 


54 | 48 | 42 | 36 | 30 | 24 | 18 | is | 24 | 30 | 36 | 42 | 48 | 54 


45 | 40 | 35 | 30 | 25 | 20 | 45 | 15 | 20 | 25 | 30 | 35 | 40 | 45 
36 | 32 | 28 | 24 | 20 | 16 12 | 16 | 20 | 24 | 28 | 32 | 36 
a7 | 24 | 21 | 18 | 15 | 12 | 09 09 | 12 | 15 | 18 | 21 | 24 | 27 


18 | 16 | 14 | 12 | 10 | 08 | 06 | | 06 | 08 | 10 | 12 | 14 | 16 | 18 


09) 08) 0'7| 06) 05) 04 03 08) 04) 05) 06} 07; O8 O9 


03 | 04 | 05 | 06 | 07 | 08 | 09 


06 | 08 | 10 | 12 | 14 | 16 | 18 


09 | 12 | 15 | 18 | 21 | 24 | 27 


12 | 16 | 20 | 24 | 28 | 32 | 36 


15 | 20 | 25 | 30 | 35 | 40 | 45 


18 | 24 | 30 | 36 | 42 | 48 | 54 


21 | 28 | 35 | 42 | 49 | 56 | 63 


24 | 32 | 40 | 48 | 56 | 64 | 72 


54 | 63 | 72 | 81 


27 | 24 | 21 | 18 | 15.| 12 | 09 


36 | 32 | 28 | 24 | 20 | 16 | 12 


) 45 | 40 | 35 | 30 | 25 | 20 


54 | 48 | 42 | 36 | 30 | 24 | 18 


63 | 56 | 49 | 42 | 35 | 28 | 21 


72 | 64 | 56 | 48 | 40 | 32 | 24 


81 | 72 | 63 | 54 | 45 | 36 | 27 


i 


HUNDREDS. TENS, 
The method of operation is as follows: the 
factors being, for example, 74x63. Two squares 
of card-board or other material are so adjusted as 


to inclose on the factor lines. 


THE COMBINATION TABLE. 91 


07 tens and 04 units horizontally, 
06 tens and 03 units perpendicularly, 


as is here represented in diagram. 


ae 


21 03 12 
“02 
07 06 05 04 08 02 O1f 01 02 03 04 
49 06 o4 


The point a being uppermost, the par- 21 
2 


tial products may be read, one in each 4212 
corner of the rectangle, and the product is 24 
perceived to be 4662. 

In a table of nine sections, the partial products 
of two 3-digit factors may be indicated by stretching 
threads squarely across the table so as to intersect 
the tens figure of each factor, and then adjusting 
the two squares to the factor figures expressing 
units and hundreds. The nine partial products 
then appear, one in each corner and one at each 


92 THE COMBINATION TABLE. 


intersection of the threads with each other, and 
with the sides of the inclosure. In the accompany- 
ing diagram are thus shown the nine partial pro- 
ducts of 361 x 427, the crossed threads being repre- 
sented by dotted lines. 


THE COMBINATION TABLE. 93 


The point a being uppermost, the par- 21 


: 7 0642 

Ep rerodacts appear as here arranged for 121907 

addition. 9402 
04 


In this diagram may also be seen an orderly 
arrangement of the partial products of many lesser 
factors, formed by dropping one or more digits from 
the original numbers. <A few of these appear below, 
and the curious student may readily discern many 


others. 
af 
301x407 Corners, 120007 
04 
2107 
301x427 Sides, 0602 
1204 
122) 
361x407 Top and Bottom, 2442 
0407 
06 
360x420 Left Lower Corner, 121200 
24 


42 
61x27 Right Upper Corner, 120 


94 THE COMBINATION TABLE. 


The Combination Table, although not adapted 
to the practical needs of the professional computer, 
may yet prove interesting and instructive to the 
philosophical student. 


CONTENTS. 


. Sections. 
Introduction, “- - - - 1-9 
The Multiplication Table, . - 10-33 
The Table of Square Numbers, . - 934-38 
The Table of Quarter-squares, : 39-43 
Multiplication as a Language, : - 44-48 

_ Spelling Backwards, - : . 49-50 
Carrying and Blundering, - sen OT 
Lightning Calculators, - - 58-61 
The Geometrical Bird, - - - 62-71 
The Double-Headed Bird, . - 72-73 
Relative Factors, - - - = i te Oo 
The Forty-inch Band, - - 76-78 
The Broad-Shouldered Bird, - 86-90 
The Headless Bird, 


Equivalent Factors, - - - 91-99 


Multiplication by Aliquot Division, 100-104 
Table of Aliquot Factors, 

Recapitulation, . - - 105-108 
Written Multiplication, = - . 109-115 
The Diamond Form, - - 116-120 


The Pyramid Form, : - 121-122 


The Inverted Pyramid Form, 


Sections. 


123-126 


Rapid Multiplication by Periods of 2 Figures 


Each, - - 
Multiplication by Subtraction, 
Brevity, : - 

Ten Examples of Brief Methods, 
Errors, - - 

Proofs Which Do Not Prove, 
Casting Out the Nines, - 
Demonstration, - - 
The Combination Table, 99x99; 


- 127 
: 128-130 
131 


133 


- 134-137 


- 138-141 
142-146 
Appendix. 


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URBANA 
17101111 


METHOD IN MENTAL MU 


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